What if you were presented with two angles that are on opposite sides of a transversal, but inside the lines? How would you describe these angles and what could you conclude about their measures? After completing this Concept, you'll be able to answer these questions using your knowledge of alternate interior angles.

Guidance

Alternate Interior Angles
are two angles that are on the
interior
of
\begin{align*}l\end{align*}
and
@$\begin{align*}m\end{align*}@$
, but on opposite sides of the transversal.
@$\begin{align*}\angle 3\end{align*}@$
and
@$\begin{align*}\angle 6\end{align*}@$
are alternate interior angles.

Alternate Interior Angles Theorem:
If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

Proof of Alternate Interior Angles Theorem:

Given
:
@$\begin{align*}l \ || \ m\end{align*}@$

Prove
:
@$\begin{align*}\angle 3 \cong \angle 6\end{align*}@$

Statement

Reason

1.
@$\begin{align*}l \ || \ m\end{align*}@$

Given

2.
@$\begin{align*}\angle 3 \cong \angle 7\end{align*}@$

Corresponding Angles Postulate

3.
@$\begin{align*}\angle 7 \cong \angle 6\end{align*}@$

Vertical Angles Theorem

4.
@$\begin{align*}\angle 3 \cong \angle 6\end{align*}@$

Transitive PoC

There are several ways we could have done this proof. For example, Step 2 could have been
@$\begin{align*}\angle 2 \cong \angle 6\end{align*}@$
for the same reason, followed by
@$\begin{align*}\angle 2 \cong \angle 3\end{align*}@$
. We could have also proved that
@$\begin{align*}\angle 4 \cong \angle 5\end{align*}@$
.

Converse of Alternate Interior Angles Theorem:
If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

Example A

Find
@$\begin{align*}m \angle 1\end{align*}@$
.

@$\begin{align*}m \angle 2 = 115^\circ\end{align*}@$
because they are corresponding angles and the lines are parallel.
@$\begin{align*}\angle 1\end{align*}@$
and
@$\begin{align*}\angle 2\end{align*}@$
are vertical angles, so
@$\begin{align*}m \angle 1 = 115^\circ\end{align*}@$
also.

@$\begin{align*}\angle 1\end{align*}@$
and the
@$\begin{align*}115^\circ\end{align*}@$
angle are alternate interior angles.

Example B

Find the measure of the angle and
@$\begin{align*}x\end{align*}@$
.

The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and solve for
@$\begin{align*}x\end{align*}@$
.